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Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.\newlinez2−18z+‾z^2 - 18z + \underline{\hspace{2em}}

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Full solution

Q. Complete the square. Fill in the number that makes the polynomial a perfect-square quadratic.\newlinez2−18z+‾z^2 - 18z + \underline{\hspace{2em}}
  1. Identify Coefficients: Identify the coefficients of the polynomial z2−18z+_z^2 - 18z + \_ to compare with the standard quadratic form ax2+bx+cax^2 + bx + c.
    a=1a = 1 (coefficient of z2z^2)
    b=−18b = -18 (coefficient of zz)
    c=?c = ? (the number we need to find)
  2. Calculate Completing Square Value: To complete the square, we need to add (b2)2(\frac{b}{2})^2 to the polynomial. In this case, bb is −18-18.\newlineCalculate (−182)2(-\frac{18}{2})^2 to find the value that completes the square.\newline(−182)2=(−9)2=81(-\frac{18}{2})^2 = (-9)^2 = 81
  3. Add to Complete Square: The number that completes the square is 8181. So the polynomial becomes a perfect-square quadratic when we add 8181. The completed square form is z2−18z+81z^2 - 18z + 81.

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